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NONSOLVABLE POLYNOMIALS WITH FIELD DISCRIMINANT 5^A

DAVID P. ROBERTS

Division of Science and Mathematics, University of Minnesota, Morris, Morris, Minnesota, 56267, USA

https://doi.org/10.1142/S1793042111004113Cited by:1 (Source: Crossref)

Abstract

We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field discriminant of the form ±p^A for p ≤ 7 a prime. Our main polynomial has degree 25, Galois group of the form PSL₂(5)⁵.10, and field discriminant 5⁶⁹. A closely related polynomial has degree 120, Galois group of the form SL₂(5)⁵.20, and field discriminant 5³¹¹. We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125 · 5^-1/12500 ≈ 124.984 and the class number of the latter field is divisible by 5⁴.

Keywords:

AMSC: 11R21, 11R29, 11S15

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